10 questions of algebra shouldnt take too long

need help answering questions

Metropolitan State University
MATH 115-52 College Algebra
Final Exam, Fall 2023
Name__________________________________
Submit the completed test to the Final Exam folder in the Assignments link located in the Assessment
drop down menu in D2L no later than 11 pm on December 9, 2023.
You can use your book, notes, calculator, D2L materials, and ML course site. You must not get help
from anyone else, real or virtual. All the work on the test should be your own. Unless directed otherwise,
show your work—there is a penalty for giving answers without justification. Good luck!
I attest that I completed this exam on my own.
Signature: ________________________________________
1.
(6 points) Just to warm up….
(a.) Simplify the algebraic expression 𝑦 2 (𝑦 −2 + 19) − 3𝑦 + 5 + 80 . Do not leave negative
exponents.
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(b.) Find the exact value of log 𝜋 (𝜋)
(c.) Find the exact value of 5log5(√2)
2. (6 points) An investment of $8,000 is made on December 9, 2015. Find the value of this
investment at the end of December 9, 2023 in each of the following situations. For full credit
show all relevant equations and give your answers accurate to two decimal places.
(a.) At an annual interest rate of 4% compounded monthly.
(b.) At an annual interest rate of 4% compounded daily.
(c.) At an annual interest rate of 4% compounded continuously.
3. (6 points) Calculate the complex number division (5 + 𝑖) ÷ (3 − 4𝑖) and write your answer in
standard complex number form 𝑎 + 𝑏𝑖. You must show at least three steps in solving this
problem.
4. Qualify the following statements as “T” (for true) or “F” (for false). You do not have to give any
reasons nor show any work. Each right answer is 1 point, each wrong answer is -1 point, and
each blank is worth 0 points.
(a.) The function 𝑓(𝑥) = 2𝑥−1 is exponential._________
(b.) For any real number 𝑥, (𝑥 + 1)2 = 𝑥 2 + 1._________
(c.) If ln(𝑥) = 5, then ln(3𝑥) = 15.________
(d.) If ln(𝑥) = 5, then ln(𝑥 3 ) = 15._________
(e.) It is possible for the graph of an exponential function to contain the points (0,5), (2,12),
(4,19). _________
(f.) If 𝑎 > 0, then log10 (1000𝑎2 ) = 3 + 2 ∗ log10 (𝑎). _________
ln⁡(𝑎)
(g.) If 𝑎, 𝑏 are positive real numbers, then ln⁡(𝑏) = ln(𝑎) − ln⁡(𝑏).________
(h.) log3 (9) = 3._________
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(i.) If log𝑏 (5) = 𝑦 and log𝑏 (3) = 𝑥, then log𝑏 (√45) = 2 𝑥 + 𝑦._________
(j.) (3 + 5𝑖) − (9 − 2𝑖) = −6 + 7𝑖.___________
5. (6 points) The following list of functions can be categorized as linear, quadratic, or exponential.
If the function is linear, find its slope and y-intercept; if it’s quadratic, identify its coefficients
(i.e., find the quadratic coefficient, the linear coefficient, and the constant coefficient); if it’s
exponential find the base and initial amount.
(a.) 𝑓(𝑥) = 3(𝑥 − 1) + 4𝑥 + 7
(b.) The points (−1,12), (1,4), and (3,12) are on the graph of the function 𝑔.
(c.) ℎ(𝑥) = 24 ∗ 4𝑥−1
6. (5 points) The graph of the function 𝑓(𝑥) = 𝑥 4 − 2𝑥 3 − 4𝑥 2 + 2𝑥 + 3 is given below. Use your
knowledge about roots of polynomials to write 𝑓 in factored form. Justify your answer.
7. (6 points) Consider the functions 𝑓(𝑥) = 𝑥 2 + 4𝑥 − 1 and 𝑔(𝑦) = 2𝑦. Calculate the following:
(a.) 𝑓(0)
(b.) (𝑔 ∘ 𝑓)(0)
𝑔
(c.) (𝑓 ) (0)
𝑓
(d.) (𝑔) (0)
(e.) (𝑓 ∘ 𝑓)(1)
(f.) (𝑓 ∘ 𝑓)(𝑥)
8. (5 points) Suppose 𝑓 is a quadratic function with vertex (−3, −5) and x-intercepts 𝑥 = −2 and
𝑥 = −4. Find its y-intercept. Make sure your answer is supported by reasons.
9. (10 points) The population of a city was 34,690 at the end of the year 2010 and growing at a rate
of 1.05% per year. Assume this annual rate of growth remains unchanged.
(a.) Give the function that determines the city’s population at time 𝑡. Define all variables used.
(b.) Find the city’s population at the end of 2023.
(c.) What is the doubling time of this city’s population?
10. (5 points) In 1986 a nuclear plant in Chernobyl, Ukraine, had a breakdown and spilled
radioactive elements into the environment. In particular, it sent the radioisotope Iodine-131,
denoted 131I, over a large area of Eastern Europe. Suppose that in a town in Western Poland the
levels of this isotope increased to 12 times the levels considered safe for living organisms. If the
half-life of 131I is 8 days, how long did it take for radioactivity levels to drop to safe levels in
this town?